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| The idealized wearing process for a convex stone, isotropic with respect to wear, can be described by an abstract Cauchy problem related to some power p of the Gauss curvature flow. We prove the existence and uniqueness of a mild solution by solving implicit Euler schemes. A free boundary can be generated due to the degeneracy of the nonlinear equation once we assume that the initial datum u_{0} has some flat region. A suitable balance between the dimension N and the power p is also required. Other contributions will be related to the shape of such worn stones. We prove that if $Np\geq 1$ then the initial flat region persists for small times under some conditions on u_{0} (finite waiting time). By means of self--similar solutions we show also that any flat region must disappear after a time large. Concerning the asymptotic behavior for large $t$, we prove that if a flat obstacle does not coincide with $u_{0}$ in an open set the same occurs in any $t>0$ (flattened retention). |
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